Quantum algorithms for searching, resampling, and hidden shift problems

Transcription

1 Quantum algorithms for searching, resampling, and hidden shift problems by Māris Ozols A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Combinatorics & Optimization Quantum Information Waterloo, Ontario, Canada, 2012 c Māris Ozols 2012

2 I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii

3 Abstract This thesis is on quantum algorithms. It has three main themes: (1) quantum walk based search algorithms, (2) quantum rejection sampling, and (3) the Boolean function hidden shift problem. The first two parts deal with generic techniques for constructing quantum algorithms, and the last part is on quantum algorithms for a specific algebraic problem. In the first part of this thesis we show how certain types of random walk search algorithms can be transformed into quantum algorithms that search quadratically faster. More formally, given a random walk on a graph with an unknown set of marked vertices, we construct a quantum walk that finds a marked vertex in a number of steps that is quadratically smaller than the hitting time of the random walk. The main idea of our approach is to interpolate the random walk from one that does not stop when a marked vertex is found to one that stops. The quantum equivalent of this procedure drives the initial superposition over all vertices to a superposition over marked vertices. We present an adiabatic as well as a circuit version of our algorithm, and apply it to the spatial search problem on the 2D grid. In the second part we study a quantum version of the problem of resampling one probability distribution to another. More formally, given query access to a black box that produces a coherent superposition of unknown quantum states with given amplitudes, the problem is to prepare a coherent superposition of the same states with different specified amplitudes. Our main result is a tight characterization of the number of queries needed for this transformation. By utilizing the symmetries of the problem, we prove a lower bound using a hybrid argument and semidefinite programming. For the matching upper bound we construct a quantum algorithm that generalizes the rejection sampling method first formalized by von Neumann in We describe quantum algorithms for the linear equations problem and quantum Metropolis sampling as applications of quantum rejection sampling. In the third part we consider a hidden shift problem for Boolean functions: given oracle access to f (x + s), where f (x) is a known Boolean function, determine the hidden shift s. We construct quantum algorithms for this problem using the pretty good measurement and quantum rejection sampling. Both algorithms use the Fourier transform and their complexity can be expressed in terms of the Fourier spectrum of f (in particular, in the second case it relates to water-filling of the spectrum). We also construct algorithms for variations of this problem where the task is to verify a given shift or extract only a single bit of information about it. iii

4 Acknowledgements First and most of all I would like to acknowledge my supervisors I was extremely lucky to have four of them! Andrew Childs and Debbie Leung supervised me at University of Waterloo, and Jérémie Roland and Martin Rötteler supervised me at NEC Laboratories America in Princeton. I would also like to acknowledge all my close friends you have taught me many lessons that are far more important than what I have learned at university. I cherish the time we have spent together it has shaped my personality in significant ways. I would not have become who I am without you all. iv

5 Table of Contents List of Tables List of Figures x xi 1 Introduction Overview Part I Part II Part III Mathematical preliminaries Semi-absorbing Markov chains and the extended hitting time Classical random walks Preliminaries Ergodicity Perron Frobenius theorem Semi-absorbing Markov chains Definition Stationary distribution Reversibility Discriminant matrix v

6 2.3.1 Definition Spectral decomposition Principal eigenvector Derivative Hitting time Definition Extended hitting time Dependence on s Adiabatic condition and the quantum hitting time of Markov chains Introduction Overview of the main result Interpolating Hamiltonian Definition Spectral decomposition The adiabatic condition The relevant subspace for adiabatic evolution The quantum adiabatic theorem Analysis of running time Choice of the schedule Running time Relation to the classical hitting time Conclusion and discussion Finding is as easy as detecting for quantum walks Introduction Related work Our approach and contributions vi

7 4.2 Preliminaries Spatial search on graphs Random walks Quantum walks Classical hitting time Quantum hitting time Discrete-time quantum walk Szegedy s construction Quantum circuit for W(s) Quantum search algorithms Algorithm with known values of p M and HT(P, M) Algorithms with approximately known p M Algorithms with a given bound on p M or HT(P, M) Application to the 2D grid Quantum rejection sampling Introduction Rejection sampling Related work Our results Definition of the problem Query complexity of quantum resampling Quantum rejection sampling algorithm Intuitive description of the algorithm Amplitude amplification subroutine and quantum rejection sampling algorithm Strong quantum rejection sampling algorithm Applications vii

8 5.5.1 Linear systems of equations Quantum Metropolis sampling Boolean function hidden shift problem Conclusion and open problems Quantum algorithms for the Boolean function hidden shift problem Introduction Hidden subgroup problem Hidden shift problem Hidden shift problem for Z d -valued functions Outline Notation and basic definitions Boolean Fourier transform Quantum Fourier transform Convolution Influence Bent functions Quantum algorithms for preparing the t-fold Fourier states Computing w s in the phase to prepare Φ t (s) Computing w s in the register to prepare Ψ t (s) Quantum algorithms for finding a hidden shift The PGM (Pretty Good Measurement) approach The Grover approach (quantum rejection sampling) The Simon approach (sampling and classical post-processing) Quantum algorithms for related problems Parity extraction Verification algorithms Zeroes in the Fourier spectrum viii

9 6.6.1 Undetectable shifts and anti-shifts Decision trees Zeroes in the t-fold Fourier spectrum Conclusions Open problems APPENDICES 157 A Water-filling vector is optimal for the SDP 158 References 162 ix

10 List of Tables 4.1 Summary of results on quantum search algorithms Reduction from quantum algorithm for linear system of equations to a quantum resampling problem Reduction from quantum Metropolis algorithm to a quantum resampling problem Summary of quantum query complexity upper bounds for the Boolean function hidden shift problem x

11 List of Figures 1.1 Structure of this thesis Markov chain P and the corresponding graph Summary of Markov chain properties Markov chain P and the corresponding absorbing chain P Rotation of v n (s) in a two-dimensional subspace Region corresponding to the double sum Extended hitting time HT(s) as a function of s Vectors U, M, and v n (s) Classical rejection sampling Classes of quantum query complexity problems Symmetrized algorithm Quantum circuit for implementing U ε Quantum algorithm for preparing the t-fold Fourier sate Φ t (s) Quantum algorithm for preparing the t-fold Fourier sate Ψ t (s) SWAP test Decision tree for function f xi

12 Chapter 1 Introduction 1.1 Overview This thesis consists of six chapters that are grouped into three parts (see Fig. 1.1). All three parts are self-contained and can be read independently from each other. The last part uses a result from the previous part, but it is not essential to know the details of this result to understand the application. 1. Introduction Part I Part II Part III 2. Markov chains [KOR10, KMOR10] 3. Quantum search (adiabatic version) [KOR10] 4. Quantum search (circuit version) [KMOR10] 5. Quantum rejection sampling [ORR12] 6. Hidden shift problem Figure 1.1: Structure of this thesis Part I The first part of this thesis is on quantizing Markov chains and is based on joint work with Hari Krovi, Frédéric Magniez, and Jérémie Roland. Most of the research for this 1

13 part was done in the summer of 2009 during my internship at NEC Laboratories America, and it was completed during subsequent short visits to Princeton. It is based on papers [KOR10, KMOR10] and consists of three chapters (see Fig. 1.1). Chapter 2 is purely classical, while Chapter 3 and Chapter 4 describe an adiabatic and a circuit-based version of a quantum walk algorithm that finds a marked vertex in a graph quadratically faster than any randomized classical algorithm. Chapter 2 contains results on Markov chains which are common to papers [KOR10] and [KMOR10]. Their proofs do not require any knowledge on quantum computing, but use only linear algebra and probability theory. The main result of this chapter is the formula p HT(s) = 2 M HT(P, M) (1.1) (1 s(1 p M )) 2 from Theorem It relates HT(P, M), the hitting time to Markov chain P, to the extended hitting time HT(s). This formula is used to analyze the quantum algorithms in Chapter 3 and Chapter 4, where it is important that HT(s) is monotonically increasing as a function of s. In Chapter 3 we describe an adiabatic quantum algorithm for finding a marked vertex in a graph that has some set M of its vertices marked. A classical version of this algorithm is a random walk with transition matrix P(s) = sp + (1 s)p, where parameter s slowly changes from s = 0 to s = 1 as the walk proceeds. During this evolution the transition matrix P(s) changes from initial matrix P to the absorbing matrix P which has no outgoing transitions from marked vertices. The main result of Chapter 3 is Theorem 3.6, which states that this algorithm finds a marked element with high probability in time O(» HT(P, M)) where HT(P, M) is the corresponding time for the classical random walk P. In Chapter 4 we use ideas from the previous chapter to design a search algorithm in the quantum circuit model, which is based on eigenvalue estimation. This algorithm also achieves a quadratic speed-up over the classical case, which is the main result of this chapter (see Theorem 4.10). We also provide several variations of this algorithm that relax the assumptions in the main theorem, as well as apply these results for the spatial search problem on the 2D grid Part II The second part of this thesis is on quantum rejection sampling. It consists of Chapter 5 and is almost identical to [ORR12], which is joint work with Martin Rötteler and Jérémie 2

14 Roland. This result originally came about as a byproduct of an algorithm for solving the Boolean function hidden shift problem, which is discussed in the third part of this thesis. I started to work on the hidden shift problem during my second internship at the NEC Laboratories America in the summer of 2010, when we discovered a Grover-like approach for solving this problem (see Sect ). The main ingredient in this approach is an amplitude amplification subroutine, which uses an oracle to implement a certain transformation between two unknown quantum states. This subroutine seemed to be of independent interest, so we decided to deviate from the original problem and study the abstract quantum state conversion problem solved by this subroutine. Only later we came to realize that it is the quantum equivalent of a simple probabilistic procedure known as rejection sampling which was studied by von Neumann [vn51]. Chapter 5 contains three main results: a query lower bound for the quantum resampling problem (see Sect. 5.3), the quantum rejection sampling algorithm (see Sect. 5.4), and applications for the linear systems of equations and quantum Metropolis sampling problems (see Sect. 5.5) Part III The third part of this thesis is based on unpublished work together with Andrew Childs, Martin Rötteler, and Jérémie Roland. It consists of Chapter 6 where we study a version of the hidden shift problem for Boolean functions. We assume that the underlying function is known and consider upper bounds on quantum query complexity by constructing quantum algorithms for this problem. The most important part of Chapter 6 is Sect. 6.4 where three different approaches for solving this problem are considered. The first approach is based on the pretty good measurement it corresponds to making all queries in parallel and performing a joint measurement of the obtained states (see Sect ). The second approach is based on quantum rejection sampling it uses amplitude amplification and performs all queries sequentially (see Sect ). The third approach is due to [Röt10, GRR11] and resembles Simon s algorithm [Sim94] it makes independent queries, each immediately followed by a measurement, and classically post-processes the obtained data (see Sect ). We also consider the problem of extracting one bit of information about the hidden shift, namely, determining the inner product with a given string (see Sect ) as well as the problem of verifying a given shift (see Sect ). The idea of using decision trees to construct Boolean functions that have large fraction of the Fourier spectrum equal to zero was suggested by Dmitry Gavinsky (see Sect ). 3

15 1.2 Mathematical preliminaries We assume that the reader is familiar with linear algebra and basics of quantum computing (for an introduction to quantum computing see [KLM07, KSV02]; for a more comprehensive overview see [NC10]). In particular, the reader should be familiar with two basic tools for constructing quantum algorithms: quantum amplitude amplification (see [KLM07, p. 163] or the original paper [BHMT00]) and eigenvalue estimation (see [KSV02, p. 125], [KLM07, p. 125.], [NC10, p. 221], or the papers [Kit95, CEMM98]). 4

16 Chapter 2 Semi-absorbing Markov chains and the extended hitting time Contents 2.1 Classical random walks Preliminaries Ergodicity Perron Frobenius theorem Semi-absorbing Markov chains Definition Stationary distribution Reversibility Discriminant matrix Definition Spectral decomposition Principal eigenvector Derivative Hitting time Definition Extended hitting time Dependence on s In this chapter we study a special type of Markov chains that can be described by a one-parameter family P(s) corresponding to convex combinations of some chain P and the corresponding absorbing chain P. Intuitively, P(s) has states that are hard to escape 5

17 which is controlled by the interpolation parameter s. For this reason we call such chains semi-absorbing. We will consider various properties of these chains as a function of the interpolation parameter s. We begin by discussing some preliminaries and defining basic concepts related to Markov chains, such as ergodicity (Sect. 2.1). Next, we define the interpolated Markov chain P(s) and consider various its properties, such as the stationary distribution and reversibility (Sect. 2.2). We proceed by applying these concepts to define and study the discriminant matrix of P(s) which encodes all relevant properties, such as eigenvalues and the principal eigenvector of P(s), but has a much more convenient form (Sect. 2.3). Finally, we define the hitting time HT and the extended hitting time HT(s) and relate the two via Theorem 2.22, which is the main result of this chapter (Sect. 2.4). Results from this chapter will be used in Chapter 3 and Chapter 4 to construct quantum algorithms based on adiabatic evolution and discrete-time quantum walks, respectively. 2.1 Classical random walks Preliminaries Let us consider a Markov chain 1 on a discrete state space X of size n. Its transition probabilities can be described by a row-stochastic matrix P, i.e., an n n matrix whose entries are real and non-negative and each row sums to one: x X : y X P xy = 1. (2.1) Here P xy is the probability to go from state x to y. A Markov chain P with state space X has a corresponding underlying directed graph with n vertices labelled by elements of X, and directed arcs labelled by non-zero probabilities P xy (see Fig. 2.1). We will represent probability distributions by row vectors whose entries are also real, non-negative, and sum to one. If p is the initial probability distribution, then the probability distribution p after one step of P is obtained by multiplying p by the transition 1 We will use terms random walk, Markov chain, and stochastic matrix interchangeably. The same holds for state, vertex, and element. 6

18 P = Ö è Figure 2.1: Markov chain P and the corresponding graph with transition probabilities. matrix from the left hand side: p = pp. A probability distribution π that satisfies πp = π is called the stationary distribution of P. For more background on Markov chains see, e.g., [GS97, KS60, KS07] Ergodicity Definition 2.1. A Markov chain is called irreducible, if any state in the underlying directed graph can be reached from any other by a finite number of steps (i.e., the graph is strongly connected); aperiodic, if there is no integer k > 1 that divides the length of every directed cycle of the underlying directed graph; ergodic, if it is both irreducible and aperiodic. Equivalently, a Markov chain P is ergodic if there exists some integer k 0 1 such that all entries of P k 0 (and, in fact, of P k for any k k 0 ) are strictly positive. Some authors call such chains regular and use the term ergodic already for irreducible chains [GS97, KS60]. From now on we will almost exclusively consider only ergodic Markov chains. Even though some of the Markov chain properties are independent from each other (such as irreducibility and aperiodicity), usually they are imposed in a specific order which is summarized in Fig As we impose more conditions, more can be said about the spectrum of P as discussed in the next section. 7

19 reversible aperiodic irreducible stochastic ergodic Figure 2.2: The order in which Markov chain properties from Definition 2.1 are typically imposed. Reversibility will be defined in Sect Perron Frobenius theorem The following theorem will be very useful for us. It is essentially the standard Perron Frobenius theorem [HJ90, Theorem 8.4.4, p. 508], but adapted for Markov chains. (This theorem is also known as the Ergodic Theorem for Markov chains [KS07, Theorem 5.9, p. 72].) The version presented here is based on the extensive overview of Perron Frobenius theory in [Mey00, Chapter 8]. Theorem (Perron Frobenius). Let P be a stochastic matrix. Then all eigenvalues of P are at most 1 in absolute value and 1 is an eigenvalue of P; if P is irreducible, then the 1-eigenvector is unique and strictly positive (i.e., it is of the form cπ for some non-vanishing probability distribution π and c = 0); if in addition to being irreducible, P is also aperiodic (i.e., P is ergodic), then the remaining eigenvalues of P are strictly smaller than 1 in absolute value. If P is irreducible but not aperiodic, it has some complex eigenvalues on the unit circle (which can be shown to be roots of unity) [Mey00, Chapter 8]. However, when in addition we also impose aperiodicity (and hence ergodicity), we are guaranteed that there is a unique eigenvalue of absolute value 1 and, in fact, it is equal to 1. 8

20 2.2 Semi-absorbing Markov chains Definition Assume that a subset M X of size m := M of the states are marked (throughout this chapter we assume that M is not empty). Let P be the Markov chain obtained from P by turning all outgoing transitions from marked states into self-loops (see Fig. 2.3). We call P the absorbing version of P (see [KS60, Chapter III] and [GS97, Sect. 11.2]). Note that P differs from P only in the rows corresponding to the marked states (where it contains all zeros on non-diagonal elements, and ones on the diagonal). If we arrange the states of X so that the unmarked states U := X \ M come first, matrices P and P have the following block structure: P := Ç PUU P MU P UM P MM å, P := Ç PUU P UM 0 I å, (2.2) where P UU and P MM are square matrices of size (n m) (n m) and m m, respectively, while P UM and P MU are matrices of size (n m) m and m (n m), respectively. U U M M Figure 2.3: Directed graphs underlying Markov chain P (left) and the corresponding absorbing chain P (right). Outgoing arcs from vertices in the marked set M have been turned into self-loops in P. Let us define an interpolated Markov chain that interpolates between P and P : P(s) := (1 s)p + sp, 0 s 1. (2.3) 9

21 This expression has some resemblance with adiabatic quantum computation where similar interpolations are usually defined for quantum Hamiltonians [FGGS00]. Indeed, we will use the interpolated Markov chain P(s) in Chapter 3 to construct an adiabatic quantum algorithm. Note that P(0) = P, P(1) = P, and P(s) has the following block structure: Ç å P P(s) = UU P UM. (2.4) (1 s)p MU (1 s)p MM + si Proposition 2.2. If P is ergodic then so is P(s) for s [0, 1). P(1) is not ergodic. Proof. A non-zero transition probability in P remains non-zero also in P(s) for s [0, 1). Thus the ergodicity of P implies that P(s) is also ergodic for s [0, 1). However, P(1) is not irreducible, since states in U are not reachable from M. Thus P(1) is not ergodic. Proposition 2.3. (P t ) UU = P t UU. Proof. Let us derive an expression for P t, the matrix of transition probabilities corresponding to t applications of P. Notice that Ä ää ä Ä a b a b = a 2 ä ab+b 0 1. By induction we get ( P t P t t 1 = UU k=0 Pk UU P ) UM. (2.5) 0 I When restricted to U, it acts as P t UU. Proposition 2.4 ([GS97, Theorem 11.3, p. 417]). If P is irreducible then lim k P k UU = 0. Intuitively this means that the sub-stochastic process defined by P UU eventually dies out or, equivalently, that the unmarked states of P eventually get absorbed (by Prop. 2.3). Proof. Let us fix an unmarked initial state x. Since P is irreducible, we can reach a marked state from x in a finite number of steps. Note that this also holds true for P. Let us denote the smallest number of steps by l x and the corresponding probability by p x. Thus in l := max x l x steps of P we are guaranteed to reach a marked state with probability at least p := min x p x > 0, independently of the initial state x U. Notice that the probability to still be in an unmarked state after l k steps is at most (1 p) k which approaches zero as we increase k. Proposition 2.5 ([KS60, Theorem 3.2.1, p. 46]). If P is irreducible then I P UU is invertible. 10

22 Proof. Notice that (I P UU ) (I + P UU + P 2 UU + + P k 1 UU ) = I Pk UU (2.6) and take the determinant of both sides. From Prop. 2.4 see that lim k det(i PUU k ) = 1. By continuity, there exists k 0 such that det(i P k 0 UU ) > 0, so the determinant of the left-hand side is non-zero as well. Using multiplicativity of determinant, we conclude that det(i P UU ) = 0 and thus I P UU is invertible. In the Markov chain literature (I P UU ) 1 is called the fundamental matrix of P Stationary distribution From now on let us demand that P is ergodic. Then according to the Perron Frobenius Theorem it has a unique and non-vanishing stationary distribution π. Let π U and π M be row vectors of length n m and m that are obtained by restricting π to sets U and M, respectively. Then π = Ä π U π M ä, π := Ä 0 U π M ä (2.7) where 0 U is the all-zeroes row vector indexed by elements of U and π satisfies π P = π. Let p M := x M π x be the probability to pick a marked element from the stationary distribution. In analogy to the definition of P(s) in Eq. (2.3), let π(s) be a convex combination of π and π, appropriately normalized: π(s) := (1 s)π + sπ (1 s) + sp M = 1 Ä ä (1 s)πu π 1 s(1 p M ) M. (2.8) Proposition 2.6. π(s) is the unique stationary distribution of P(s) for s [0, 1). At s = 1 any distribution with support only on marked states is stationary, including π(1). Proof. Notice that (π π )(P P ) = Ä π U 0 ä Ç å 0 0 = 0 (2.9) P MU P MM I which is equivalent to πp + π P = πp + π P. (2.10) 11

23 Using this equation we can check that π(s)p(s) = π(s) for any s [0, 1]: Ä (1 s)π + sπ ää (1 s)p + sp ä (2.11) = (1 s) 2 πp + (1 s)s(πp + π P) + s 2 π P (2.12) = (1 s) 2 π + (1 s)s(π + π ) + s 2 π (2.13) = Ä (1 s)π + sπ ää (1 s) + s ä (2.14) = (1 s)π + sπ. (2.15) Recall from Prop. 2.2 that P(s) is ergodic for s [0, 1) so π(s) is the unique stationary distribution by Perron Frobenius Theorem. Since P acts trivially on marked states, any distribution with support only on marked states is stationary for P(1) Reversibility Definition 2.7. Markov chain P is called reversible if it is ergodic and satisfies the socalled detailed balance condition where π is the unique stationary distribution of P. x, y X : π x P xy = π y P yx (2.16) Intuitively this means that the net flow of probability in the stationary distribution between every pair of states is zero. Note that Eq. (2.16) is equivalent to diag(π) P = P T diag(π) = Ä diag(π)p ä T (2.17) where diag(π) is a diagonal matrix whose diagonal is given by vector π. Thus Eq. (2.16) is equivalent to saying that matrix diag(π)p is symmetric. Proposition 2.8. If P is reversible then so is P(s) for any s [0, 1]. Hence, P(s) satisfies the extended detailed balance equation s [0, 1], x, y X : π x (s)p xy (s) = π y (s)p yx (s). (2.18) Proof. First, notice that the absorbing walk P is reversible 2 since Ç å Ç å Ç å diag(π )P 0 0 PUU P = UM 0 0 = = diag(π 0 diag(π M ) 0 I 0 diag(π M ) ) (2.19) 2 Strictly speaking, the definition of reversibility also includes ergodicity for the stationary distribution to be uniquely defined. However, we will relax this requirement for P since, by continuity, π is the natural choice of the unique stationary distribution. 12

24 which is symmetric. Next, notice that diag(π π )(P P ) = which gives us an analogue of Eq. (2.10): Ç å Ç å diag(πu ) = 0 (2.20) 0 0 P MU P MM I diag(π )P + diag(π)p = diag(π)p + diag(π )P. (2.21) Here the right-hand side is symmetric due to reversibility of P and P, thus so is the left-hand side. Using this we can check that P(s) is reversible: diag Ä (1 s)π + sπ ää (1 s)p + sp ä (2.22) = (1 s) 2 diag(π)p + (1 s)s Ä diag(π)p + diag(π )P ä + s 2 diag(π )P (2.23) where the first and last terms are symmetric since P and P are reversible, but the middle term is symmetric due to Eq. (2.21). 2.3 Discriminant matrix Definition The discriminant matrix of a Markov chain P(s) is D(s) :=» P(s) P(s) T, (2.24) where the Hadamard product and the square root are computed entry-wise. This matrix was introduced by Szegedy in [Sze04a, Sze04b]. We prefer to work with D(s) rather than P(s) since the matrix of transition probabilities is not necessarily symmetric while its discriminant matrix is. Proposition 2.9. If P is reversible then D(s) = diag Ä» π(s) ä P(s) diag Ä» π(s) ä 1, s [0, 1); (2.25) D(1) = ( Ä ä diag πu PUU diag Ä ä 1 ) πu 0. (2.26) 0 I 13

25 Here the square roots are also computed entry-wise and M 1 denotes the matrix inverse of M. Notice that for s [0, 1) the right-hand side of Eq. (2.25) is well-defined, since P(s) is ergodic by Prop. 2.2 and thus according to the Perron Frobenius Theorem has a unique and non-vanishing stationary distribution. However, recall from Prop. 2.6 that π(1) vanishes on U, so the right-hand side of Eq. (2.25) is no longer well-defined at s = 1. For this reason we have an alternative expression for D(1). Proof (of Prop. 2.9). For a reversible Markov chain P the extended detailed balance condition in Eq. (2.18) implies that D xy (s) =» P xy (s)p yx (s) = P xy (s)» π x (s)/π y (s). This is equivalent to Eq. (2.25). At s = 1 from Eq. (2.24) we have: D(1) =» P(1) P(1) T = Ã ÇPUU PUU T 0 å = 0 I (» PUU PUU T 0 ). (2.27) 0 I In the same way one can use Eq. (2.24) to compute the upper left block of D(s) for any s, and notice that it does not depend on s: D UU (s) = P UU P T UU = D UU(0) = diag Ä πu ä PUU diag Ä πu ä 1 (2.28) where the last equality follows from Eq. (2.25) at s = 0. Together with Eq. (2.27) this gives us the desired expression in Eq. (2.26) Spectral decomposition Recall from Eq. (2.24) that D(s) is real and symmetric. Therefore, its eigenvalues are real and it has an orthonormal set of real eigenvectors. Let D(s) = n λ i (s) v i (s) v i (s) (2.29) i=1 be the spectral decomposition of D(s) with eigenvalues λ i (s) and eigenvectors 3 v i (s). Moreover, let us arrange the eigenvalues so that λ 1 (s) λ 2 (s) λ n (s). (2.30) 3 There is no need to use bra-ket notation at this point; nevertheless we adopt it since vectors v i (s) later will be used as quantum states. 14

26 From now on we will assume that P is reversible (and hence ergodic) without explicitly mentioning it. Under this assumption the matrices P(s) and D(s) are similar (see Prop below). This means that D(s) essentially has the same properties as P(s), but in addition it also admits a spectral decomposition with orthogonal eigenvectors. This will be very useful in Chapter 3, where we will find the spectral decomposition of Hamiltonian H(s) in terms of that of D(s), and use it to relate properties of H(s) and P(s). Proposition The matrices P(s) and D(s) are similar for any s [0, 1] and therefore have the same eigenvalues. In particular, the eigenvalues of P(s) are real. Proof. From Eq. (2.25) we see that the matrices D(s) and P(s) are similar for s [0, 1). From Eq. (2.26) we see that D(1) is similar to P := Ä ä P UU 0 0 I. To verify that P and P(1) = Ä ä Ä ä PUU P UM 0 I are similar, let M := PUU I P UM 0 I. One can check that MP(1)M 1 = P where M 1 = Ä (P UU I) 1 (P UU I) 1 ä P UM exists, since PUU I is invertible according 0 I to Prop By transitivity, D(1) is also similar to P(1). Proposition The largest eigenvalue of D(s) is 1. It has multiplicity 1 when s [0, 1) and multiplicity m when s = 1. In other words, λ n 1 (s) < λ n (s) = 1, s [0, 1), (2.31) λ n m (1) < λ n m+1 (1) = = λ n (1) = 1. (2.32) Proof. Let us argue about P(s), since it has the same eigenvalues as D(s) by Prop From the Perron Frobenius Theorem we have that i : λ i (s) 1 and λ n (s) = 1. In addition, by Prop. 2.2 the Markov chain P(s) is ergodic for any s [0, 1), so i = n : λ i (s) < 1. Finally, note by Eq. (2.26) that for s = 1 eigenvalue 1 has multiplicity at least m. Recall from Eq. (2.28) that D UU (1) and P UU are similar. From Prop. 2.5 we conclude that all eigenvalues of P UU are strictly less than 1. Thus the multiplicity of eigenvalue 1 of D(1) is exactly m Principal eigenvector Let us prove an analogue of Prop. 2.6 for the matrix D(s). Proposition 2.12.» π(s) T is the unique (+1)-eigenvector of D(s) for s [0, 1). At s = 1 any vector with support only on marked states is a (+1)-eigenvector, including» π(1) T. 15

27 Proof. Since P(s) is row-stochastic, P(s) 1 T X = 1T X where 1 X is the all-ones row vector. Thus we can check that for s [0, 1), D(s)» Å»π(s) ã»π(s) ã 1»π(s) π(s) T = diag P(s) diagå T (2.33) Å»π(s) ã = diag P(s) 1 T X (2.34) Å»π(s) ã = diag 1 T X (2.35) =» π(s) T. (2.36) Uniqueness for s [0, 1) follows by the uniqueness of π(s) and Prop For the s = 1 case, notice from Eq. (2.26) that D(1) acts trivially on marked elements and recall from Eq. (2.8) that π(1) = (0 U π M )/p M. According to the above Proposition, for any s [0, 1] we can choose the principal eigenvector v n (s) in the spectral decomposition of D(s) in Eq. (2.29) to be v n (s) :=» π(s) T. (2.37) We would like to have an intuitive understanding of how v n (s) evolves as a function of s. Let us introduce some useful notation that we will also need later. Let 0 U and 1 U (respectively, 0 M and 1 M ) be the all-zeros and all-ones row vectors of dimension n m (respectively, m) whose entries are indexed by elements of U (respectively, M). Furthermore, let π U := π U /(1 p M ), π M := π M /p M (2.38) be the normalized row vectors describing the stationary distribution π restricted to unmarked and marked states. Let us also define the following unit vectors in R n : U :=» ( π U 0 M ) T = M :=» (0 U π M ) T = 1 pm 1» 1 pm x U x M Then we can express v n (s) as a linear combination of U and M. πx x, (2.39) πx x. (2.40) 16

28 Proposition v n (s) = cos θ(s) U + sin θ(s) M where cos θ(s) = Ã (1 s)(1 p M ) p, sin θ(s) = M 1 s(1 p M ) 1 s(1 p M ). (2.41) Proof. By substituting π(s) from Eq. (2.8) into Eq. (2.37) we get v n (s) =» π(s) T = Õ Ä(1 s)πu π M ä T 1 s(1 p M ) Õ Ä(1 ä T s)(1 pm ) π U p M π M = 1 s(1 p M ) (2.42) which is the desired expression. Thus v n (s) lies in the two-dimensional subspace span{ U, M } and is subject to a rotation as we change the parameter s (see Fig. 2.4). In particular, v n (0) =» 1 p M U + p M M, v n (1) = M. (2.43) M = v n (1) v n (0) Figure 2.4: As s changes from zero to one, the evolution of the principal eigenvector v n (s) corresponds to a rotation in the two-dimensional subspace span{ U, M }. U Proposition θ(s) and its derivative θ(s) := ds d θ(s) are related as follows: 2 θ(s) = sin θ(s) cos θ(s). (2.44) 1 s Proof. Notice that d Ä sin 2 θ(s) ä = 2 θ(s) sin θ(s) cos θ(s). (2.45) ds 17

29 On the other hand, according to Eq. (2.41) we have d Ä sin 2 θ(s) ä = d ( ) p M ds ds 1 s(1 p M ) By comparing both equations we get the desired result. = p M(1 p M ) (1 s(1 p M )) 2 = sin2 θ(s) cos 2 θ(s). (2.46) 1 s Derivative Proposition D(s) and its derivative Ḋ(s) := ds d D(s) are related as follows: Ḋ(s) = 1 ΠM, I D(s) (2.47) 2(1 s) where {X, Y} := XY + YX is the anticommutator of X and Y, and Π M := x M x x is the projector onto the m-dimensional subspace spanned by marked states M. Proof. Recall from Eq. (2.24) that D(s) =» P(s) P(s) T. The block structure of P(s) is given in Eq. (2.4). First, let us derive an expression for D MM (s), the lower right block of D(s): D MM (s) =» P MM (s) P MM (s) T (2.48) = Ä (1 s)pmm + si ä Ä (1 s)p T MM + siä. (2.49) Let us separately consider the diagonal and off-diagonal entries of D MM (s). For x, y M we have (1 s)» P xy P yx if x = y, D xy (s) = (2.50) (1 s)p xx + s if x = y. Thus we can write D MM (s) as D MM (s) = (1 s) P MM PMM T + si. (2.51) Expressions for the remaining blocks of D(s) can be derived in a straightforward way. By putting all blocks together we get (» PUU P D(s) = UU T» (1 s)(pum P» MU T ) ) (1 s)(pmu PUM T ) (1 s)» P MM PMM T + si. (2.52) 18

30 When we take the derivative with respect to s we find Ḋ(s) = Ñ 0 1» 2 PUM P T 1 s MU 1» 2 PMU P T 1 s UM I» P MM PMM T é. (2.53) To relate Ḋ(s) and the original matrix D(s), observe that (» 0 (1 s)(pum P Π M D(s) + D(s)Π M =» MU T ) ) (1 s)(pmu PUM T ) 2(1 s)» P MM PMM T + 2sI (2.54) which can be seen by overlaying the second column and row of D(s) given in Eq. (2.52). When we rescale this by an appropriate constant, we get Ñ 1 2(1 s) {Π M, D(s)} = s 0 1» 2 PUM P T» 1 s MU PMU PUM T» P MM PMM T 1 s s I é. (2.55) This is very similar to the expression for Ḋ(s) in Eq. (2.53), except for a slightly different coefficient for the identity matrix in the lower right corner. We can correct this by adding Π M with an appropriate constant: 2(1 s) 1 {Π M, D(s)} + 1 s 1 Π M = Ḋ(s). 2.4 Hitting time Definition To define the hitting time of Markov chain P, let us consider a simple classical algorithm for finding a marked element in the state space X using a random walk based on P. Random Walk Algorithm 1. Sample a vertex x X according to the stationary distribution π of P. 2. If x is marked, output x and exit. 3. Otherwise, update x according to P and go back to step 2. The hitting time of P is the expected number of applications of P during this algorithm (notice that the algorithm stops as soon as a marked element is reached, thus effectively it uses the absorbing Markov chain P ). Here is a more formal definition: 19

31 Definition Let P be an ergodic Markov chain, and M be a set of marked states. The hitting time of P with respect to M, denoted by HT(P, M), is the expected number of executions of the last step of the Random Walk Algorithm, conditioned on the initial vertex being unmarked. Proposition The hitting time of Markov chain P with respect to marked set M is HT(P, M) = t=0 U D t (1) U. (2.56) Proof. The expected number of iterations in the Random Walk Algorithm is HT(P, M) := = = = = l Pr[need exactly l steps] (2.57) l=1 l Pr[need exactly l steps] (2.58) l=1 t=1 Pr[need exactly l steps] (2.59) t=1 l=t Pr[need at least t steps] (2.60) t=1 Pr[need more than t steps] (2.61) t=0 where the region corresponding to the double sum is shown in Fig t l Figure 2.5: The region corresponding to the double sum in Eqs. (2.58) and (2.59). 20

32 Hence, we can consider the probability that no marked vertex is found after t steps starting from an unmarked vertex distributed according to π U = π U /(1 p M ). Let us find an explicit expression for this probability. The distribution of vertices at the first execution of step 3 of the Random Walk Algorithm is ( π U 0 M ), hence Pr[need more than t steps] = ( π U 0 M )P t (1 U 0 M ) T. (2.62) Recall from Prop. 2.3 that (P t ) UU = PUU t so we can simplify Eq. (2.62) as follows: Pr[need more than t steps] = ( π U 0 M )P t (1 U 0 M ) T (2.63) = π U 1 p M P t UU1 T U (2.64) = πu 1 p M diag Ä πu ä P t UU diag Ä πu ä 1 π T U 1 p M (2.65) = U D t (1) U, (2.66) where the last equality follows from the expression for the discriminant matrix D(1) in Eq. (2.26). By plugging this back in Eq. (2.61) we get the desired result Extended hitting time Let us define the following extension of the hitting time based on Eq. (2.56): HT(s) := U Ä D t (s) v n (s) v n (s) ä U. (2.67) t=0 Note that HT(1) = HT(P, M) since U v n (1) = U M = 0. This justifies calling HT(s) extended hitting time. We can use the similarity transformation between D(s) and P(s) from Prop. 2.9 to obtain an alternative expression for HT(s): HT(s) = ( π U 0 M ) Ä P t (s) Q(s) ä (1 U 0 M ) T (2.68) t=0 where Q(s) := lim t P t (s) is a stochastic matrix whose all rows are equal to π(s). Intuitively, HT(s) may be understood as the time it takes for P(s) to converge to its stationary distribution π(s), starting from ( π U 0 M ). For s = 1, the walk P(1) = P converges to the (non-unique) stationary distribution (0 U π M ), which only has support over marked elements. 21

33 Proposition The extended hitting time can be expressed as HT(s) = U A(s) U, A(s) := k: λ k (s) =1 v k (s) v k (s). (2.69) 1 λ k (s) Proof. Rewrite Eq. (2.67) using the spectral decomposition of D(s) from Eq. (2.29): HT(s) = t=0 k =n λ t k (s) U v k(s) v k (s) U = k: λ k (s) =1 v k (s) U 2 1 λ k (s) (2.70) where we exchanged the sums and used the expansion (1 x) 1 = t=0 x t. For technical reasons it will be important later that all eigenvalues of P(s) are nonnegative. We can guarantee this using a standard trick we replace the original Markov chain P with the lazy walk (P + I)/2 where I is the n n identity matrix. In fact, we can assume without loss of generality that the original Markov chain already is lazy, since this affects the hitting time only by a constant factor, as shown below. Proposition Let P be an ergodic and reversible Markov chain. Then for any s [0, 1] the eigenvalues of (P(s) + I)/2 are between 0 and 1. Moreover, if the extended hitting time of P is HT(s), then the extended hitting time of (P + I)/2 is 2 HT(s). Proof. Since P is reversible, so is P(s) by Prop Thus the eigenvalues of P(s) are real by Prop If λ k (s) is an eigenvalue of P(s) then λ k (s) [ 1, 1] according to Perron Frobenius Theorem. Thus, the eigenvalues of (P(s) + I)/2 satisfy (λ k (s) + 1)/2 [0, 1]. Recall from Prop that P(s) and D(s) are similar. Thus, the discriminant matrix of (P(s) + I)/2 is (D(s) + I)/2, which has the same eigenvectors as D(s). By Prop we see that the extended hitting time of (P(s) + I)/2 is given by k: λ k (s) =1 v k (s) U 2 1 λ. (2.71) k(s)+1 2 Since 1 λ k(s)+1 2 = 1 λ k(s) 2, the above expression is equal to 2 HT(s) as claimed. The following property of A(s) will be useful on several occasions. cos θ(s) Proposition A(s) M = sin θ(s) A(s) U. Proof. Recall from Prop that λ n (s) = 1, so A(s) v n (s) = 0 by definition. If we substitute v n (s) = cos θ(s) U + sin θ(s) M from Prop in this equation, we get the desired formula. 22

34 2.4.3 Dependence on s The goal of this section is to express HT(s) as a function of s and the hitting time HT(P, M) of the original Markov chain. The main idea is to relate ds d HT(s) to HT(s) and then solve the resulting differential equation. Lemma The derivative of HT(s) is related to HT(s) as d ds HT(s) = 2(1 p M) HT(s) (2.72) 1 s(1 p M ) where p M is the probability to pick a marked state from the stationary distribution π of P. Proof. Recall from Prop that HT(s) = U A(s) U where A(s) may be written as A(s) = B(s) 1 Π n (s) where B(s) := I D(s) + Π n (s), Π n (s) := v n (s) v n (s). (2.73) Recall from Sect that v n (s) is the unique (+1)-eigenvector of D(s) for s [0, 1), thus B(s) is indeed invertible when s is in this range. From now on we will not write the dependence on s explicitly. We will also often use f (s) as a shorthand form of ds d f (s). Let us start with d HT = U Ȧ U (2.74) ds and expand Ȧ using Eq. (2.73). To find ds d (B 1 ), take the derivative of both sides of B 1 B = I and get ds d (B 1 ) B + B 1 dds B = 0. Thus ds d (B 1 ) = B 1 ḂB 1 and Ȧ = B 1 ḂB 1 Π n. (2.75) Notice from Eq. (2.73) that Ḃ = Ḋ + Π n, thus Ȧ = B 1 ( Ḋ + Π n )B 1 Π n and d ds HT = h 1 + h 2 + h 3 where Let us evaluate each of these terms separately. h 1 := U B 1 ḊB 1 U, (2.76) h 2 := U B 1 Π n B 1 U, (2.77) h 3 := U Π n U. (2.78) 23

35 To evaluate the first term h 1, we substitute Ḋ = 1 and replace I D by B Π n according to Eq. (2.73): 2(1 s) ΠM, I D from Prop (1 s)h 1 = U B 1 {Π M, B Π n }B 1 U (2.79) = U B 1Ä {Π M, B} {Π M, Π n } ä B 1 U (2.80) = U {B 1, Π M } U U B 1 {Π M, Π n }B 1 U. (2.81) Recall that Π M = x M x x is the projector onto the marked states. Thus Π M U = 0 and the first term vanishes. Note that B has the same eigenvectors as D. In particular, B 1 v n = v n and thus B 1 Π n = Π n = Π n B 1. Using this we can expand the anticommutator in the second term: B 1 {Π M, Π n }B 1 = B 1 Π M Π n + Π n Π M B 1. Since all three matrices in this expression are real and symmetric and U is also real, both terms of the anti-commutator have the same contribution, so we get 2(1 s)h 1 = 2 U B 1 Π M Π n U. (2.82) Recall from Prop that v n = cos θ U + sin θ M, so we see that Π M Π n U = Π M v n v n U = sin θ M cos θ. Moreover, B 1 = A + Π n according to Eq. (2.73), so 2(1 s)h 1 = 2 sin θ cos θ U (A + Π n ) M. (2.83) Recall from Prop that sin θ U A M = cos θ U A U. To simplify the second term, notice that U Π n M = U v n v n M = cos θ sin θ. When we put this together, we get 2(1 s)h 1 = 2 cos 2 θ U A U 2 sin 2 θ cos 2 θ (2.84) or simply h 1 = cos2 θ Ä U A U sin 2 θ ä. (2.85) 1 s Let us now consider the second term h 2 = U B 1 Π n B 1 U. First, we compute Π n = v n v n + v n v n. Using B 1 v n = v n we get B 1 Π n B 1 = B 1 v n v n + v n v n B 1. Since v n U = cos θ we have h 2 = 2 U B 1 v n cos θ (2.86) where the factor two comes from the fact that all vectors involved are real and matrix B 1 is real and symmetric. Let us compute v n = θ Ä sin θ U + cos θ M ä. (2.87) 24

36 Notice that v n v n = 0 and thus Π n v n = 0. By substituting B 1 = A + Π n from Eq. (2.73) we get h 2 = 2 U A v n cos θ. (2.88) Next, we substitute v n and get h 2 = 2 θ Ä sin θ U A U + cos θ U A M ä cos θ. (2.89) Now we use Prop to substitute A M by A U : h 2 = 2 θ ( ) sin θ cos2 θ U A U cos θ = 2 θ cos θ U A U. (2.90) sin θ sin θ Finally, we substitute 2 θ = sin θ cos θ 1 s from Eq. (2.44) and get h 2 = cos2 θ U A U. (2.91) 1 s For the last term h 3 = U Π n U we observe that U v n v n U = θ sin θ cos θ thus h 3 = 2 θ sin θ cos θ where the factor two comes from symmetry. After substituting 2 θ from Eq. (2.44) we get h 3 = cos2 θ 1 s sin2 θ. (2.92) When we compare Eqs. (2.85), (2.91), and (2.92) we notice that h 2 = h 1 + h 3. Thus the derivative of the hitting time is d ds HT = h 1 + h 2 + h 3 = 2h 2. Recall from Eq. (2.69) that HT = U A U. Thus d ds HT(s) = θ(s) 2cos2 HT(s). (2.93) 1 s By substituting cos θ(s) from Eq. (2.41) we get the desired result. Theorem The extended hitting time HT(s) is related to HT(P, M), the hitting time of Markov chain P with marked states M, as follows: HT(s) = p 2 M HT(P, M) (2.94) (1 s(1 p M )) 2 where p M is the probability to pick a marked state from the stationary distribution π of P. 25

37 Proof. We will prove this theorem by solving the differential equation from Lemma sin θ cos θ In particular, let us consider Eq. (2.93). Recall from Eq. (2.44) that 2 θ = 1 s, so we can rewrite the coefficient in this equation as 2 cos2 θ 1 s = 2 sin θ cos θ 1 s Now we can rewrite the differential equation as By integrating both sides we get d ds HT(s) HT(s) cos θ sin θ = 4 θ cos θ d sin θ = 4 ds (sin θ) sin θ = 4 d ds. (2.95) (sin θ(s)). (2.96) sin θ(s) ln HT(s) = 4 ln sin θ(s) + C (2.97) for some constant C. Recall from Sect that HT(1) = HT(P, M) and from Eq. (2.41) that sin θ(1) = 1, so the boundary condition at s = 1 gives us C = ln HT(P, M). Since all quantities are non-negative, we can omit the absolute value signs. After exponentiating both sides we get HT(s) = sin 4 θ(s) HT(P, M). (2.98) We get the desired expression when we substitute sin θ(s) from Eq. (2.41). In the next two chapters we consider several quantum search algorithms whose running time depends on HT(s) for some values of s [0, 1]. Theorem 2.22 is a crucial ingredient in analysis of these algorithms, since it relates HT(s) to the usual hitting time HT(P, M). In particular, it is important that HT(s) is monotonically increasing as a function of s (some example plots of HT(s) are shown in Fig. 2.6). 26